The Earth-Kzin Wars were a series of four conflicts between Earth forces and the Kzinti in the mid-21st century. The last war was fought in 2069.
Although the Kzinti had opportunities to taste "Human meat" during these conflicts, ultimately the wars ended badly for the Kzinti with a decisive victory for Earth.
In the end, Earth imposed many restrictions upon the Kzinti, through the provisions made by the Treaty of Sirius. These restrictions included the limiting of their space fleet to a few police vessels. (TAS: "The Slaver Weapon")
In Larry Niven's Known Space universe where the Kzin originated from, they were known as the Man-Kzin Wars.
Given the timeframe, this is possibly the earliest major conflict between Earth and another race.